Collapsed Riemannian and non-Riemannian spaces
We will present a topic course that covers on the collapsing theory in Metric Riemannian geometry. We will start with the Gromov’s almost flat manifolds, the nilpotent structure theory of Cheeger-Fukaya-Gromov. We will discuss similar nilpotent structures found on collapsed Riemannian manifolds with local Ricci bounded covering geometry, and on Alexandrov spaces (non-Riemannian spaces) with local bounded covering geometry.
CONTENTS
I. Collapsed Manifolds with Bounded Sectional Curvature
1.1. Examples 1.2. The almost flat manifolds 1.3. The nilpotent fiber bundle theorem 1.4. The singular nilpotent fibration theorem 1.5. Applications
II. Collapsed Manifolds with Local Bounded Ricci Covering Geometry
2.1. Examples 2.2. The Cheeger-Colding-Naber theory on Ricci limit spaces 2.3. The Margulis lemma 2.4. Maximally collapsed manifolds with local bounded Ricci covering geometry 2.5. The nilpotent fibration theorem 2.6. Applications
III. Collapsed Alexandrov Spaces
3.1. Basic Alexandrov spaces 3.2. The fibration theorem 3.3. Maximally collapsed Alexandrov space with local bounded covering geometry 3.4. The nilpotent fibration theorem 3.5. Singular fibration structures 3.6. Applications
Reference: Notes on convergence and collapsing theorems in Riemannian geometry, Handbook of Geometric Analysis (Vol II) ALM 13, pp. 193-298. Typeset by AMS
Notes on convergence and collapsing theorems in Riemannian geometry, Handbook of Geometric Analysis (Vol II) ALM 13, pp. 193-298.
Riemannian geometry